Membrane capacitive memory alters spiking in neurons described by the fractional-order Hodgkin-Huxley model.

Weinberg SH - PLoS ONE (2015)

Bottom Line:
We find that in the membrane patch model, as fractional-order decreases, i.e., a greater influence of membrane potential memory, peak sodium and potassium currents are altered, and spike frequency and amplitude are generally reduced.In the nerve axon, the velocity of spike propagation increases as fractional-order decreases, while in a neural network, electrical activity is more likely to cease for smaller fractional-order.Importantly, we demonstrate that the modulation of the peak ionic currents that occurs for reduced fractional-order alone fails to reproduce many of the key alterations in spiking properties, suggesting that membrane capacitive memory and fractional-order membrane potential dynamics are important and necessary to reproduce neuronal electrical activity.

ABSTRACTExcitable cells and cell membranes are often modeled by the simple yet elegant parallel resistor-capacitor circuit. However, studies have shown that the passive properties of membranes may be more appropriately modeled with a non-ideal capacitor, in which the current-voltage relationship is given by a fractional-order derivative. Fractional-order membrane potential dynamics introduce capacitive memory effects, i.e., dynamics are influenced by a weighted sum of the membrane potential prior history. However, it is not clear to what extent fractional-order dynamics may alter the properties of active excitable cells. In this study, we investigate the spiking properties of the neuronal membrane patch, nerve axon, and neural networks described by the fractional-order Hodgkin-Huxley neuron model. We find that in the membrane patch model, as fractional-order decreases, i.e., a greater influence of membrane potential memory, peak sodium and potassium currents are altered, and spike frequency and amplitude are generally reduced. In the nerve axon, the velocity of spike propagation increases as fractional-order decreases, while in a neural network, electrical activity is more likely to cease for smaller fractional-order. Importantly, we demonstrate that the modulation of the peak ionic currents that occurs for reduced fractional-order alone fails to reproduce many of the key alterations in spiking properties, suggesting that membrane capacitive memory and fractional-order membrane potential dynamics are important and necessary to reproduce neuronal electrical activity.

pone.0126629.g007: Sub-threshold impulse and voltage response in the passive fractional-order cable equation.(A) The impulse response function G(x/λ, t/τ) is shown as a function of space x, normalized by space constant λ, at times t = 0.05τ and t = τ, where τ is the time constant, on a linear (top) and logarithmic (bottom) scale, for different values of fractional-order α. (B) G(x/λ, t/τ) is shown as a function of normalized time t/τ at location x = 0 (top) and x = λ (bottom). (C) The normalized voltage response to a current step input at the origin x = 0 is shown as a function of normalized time t/τ at locations x = 0 and x = λ. The voltage response in the membrane patch is shown for comparison (dashed lines, Fig 1C). (D) The normalized position of stimulus propagation x/λ is shown as a function of normalized time t/τ (the time at which the normalized voltage response is 0.5) (E) The pseudo-velocity, given by the slope of the stimulus propagation, in units of λ/τ, is shown as a function of α.

Mentions:
As in the previous section, we first investigate the properties of the passive membrane. The fractional-order passive cable equation is given by Eq 6 with the additional of a voltage diffusion term:Cmα∂αVm∂tα+1RmVm=g∂2Vm∂x2+I(x,t),(19)where g is a longitudinal cable conductance. Eq 19 can be written in standard form,τα∂αVm∂tα=λ2∂2Vm∂x2−Vm+RmI(x,t),(20)where time constant and space constant . The membrane potential Vm(x, t) can be determined byVm(x,t)=Rm∫0t∫−∞∞G(x−x′,t−t′)I(x′,t′)dx′dt′,(21)the convolution of the applied current I(x, t) and G(x, t), the impulse response, scaled by Rm. We solve for G(x, t) using an analytical-numerical approach using the Laplace-Fourier transform (see S1 Text), shown in Fig 7A and 7B. At early time points, for small α, G(x, t) is more “spread out” in space, while G(x, t) is less spread out at later time points (Fig 7A). At the site of the impulse (x = 0) and one length constant away (x = λ), the impulse is also more spread out in time for small α (Fig 7B).

pone.0126629.g007: Sub-threshold impulse and voltage response in the passive fractional-order cable equation.(A) The impulse response function G(x/λ, t/τ) is shown as a function of space x, normalized by space constant λ, at times t = 0.05τ and t = τ, where τ is the time constant, on a linear (top) and logarithmic (bottom) scale, for different values of fractional-order α. (B) G(x/λ, t/τ) is shown as a function of normalized time t/τ at location x = 0 (top) and x = λ (bottom). (C) The normalized voltage response to a current step input at the origin x = 0 is shown as a function of normalized time t/τ at locations x = 0 and x = λ. The voltage response in the membrane patch is shown for comparison (dashed lines, Fig 1C). (D) The normalized position of stimulus propagation x/λ is shown as a function of normalized time t/τ (the time at which the normalized voltage response is 0.5) (E) The pseudo-velocity, given by the slope of the stimulus propagation, in units of λ/τ, is shown as a function of α.

Mentions:
As in the previous section, we first investigate the properties of the passive membrane. The fractional-order passive cable equation is given by Eq 6 with the additional of a voltage diffusion term:Cmα∂αVm∂tα+1RmVm=g∂2Vm∂x2+I(x,t),(19)where g is a longitudinal cable conductance. Eq 19 can be written in standard form,τα∂αVm∂tα=λ2∂2Vm∂x2−Vm+RmI(x,t),(20)where time constant and space constant . The membrane potential Vm(x, t) can be determined byVm(x,t)=Rm∫0t∫−∞∞G(x−x′,t−t′)I(x′,t′)dx′dt′,(21)the convolution of the applied current I(x, t) and G(x, t), the impulse response, scaled by Rm. We solve for G(x, t) using an analytical-numerical approach using the Laplace-Fourier transform (see S1 Text), shown in Fig 7A and 7B. At early time points, for small α, G(x, t) is more “spread out” in space, while G(x, t) is less spread out at later time points (Fig 7A). At the site of the impulse (x = 0) and one length constant away (x = λ), the impulse is also more spread out in time for small α (Fig 7B).

Bottom Line:
We find that in the membrane patch model, as fractional-order decreases, i.e., a greater influence of membrane potential memory, peak sodium and potassium currents are altered, and spike frequency and amplitude are generally reduced.In the nerve axon, the velocity of spike propagation increases as fractional-order decreases, while in a neural network, electrical activity is more likely to cease for smaller fractional-order.Importantly, we demonstrate that the modulation of the peak ionic currents that occurs for reduced fractional-order alone fails to reproduce many of the key alterations in spiking properties, suggesting that membrane capacitive memory and fractional-order membrane potential dynamics are important and necessary to reproduce neuronal electrical activity.

ABSTRACTExcitable cells and cell membranes are often modeled by the simple yet elegant parallel resistor-capacitor circuit. However, studies have shown that the passive properties of membranes may be more appropriately modeled with a non-ideal capacitor, in which the current-voltage relationship is given by a fractional-order derivative. Fractional-order membrane potential dynamics introduce capacitive memory effects, i.e., dynamics are influenced by a weighted sum of the membrane potential prior history. However, it is not clear to what extent fractional-order dynamics may alter the properties of active excitable cells. In this study, we investigate the spiking properties of the neuronal membrane patch, nerve axon, and neural networks described by the fractional-order Hodgkin-Huxley neuron model. We find that in the membrane patch model, as fractional-order decreases, i.e., a greater influence of membrane potential memory, peak sodium and potassium currents are altered, and spike frequency and amplitude are generally reduced. In the nerve axon, the velocity of spike propagation increases as fractional-order decreases, while in a neural network, electrical activity is more likely to cease for smaller fractional-order. Importantly, we demonstrate that the modulation of the peak ionic currents that occurs for reduced fractional-order alone fails to reproduce many of the key alterations in spiking properties, suggesting that membrane capacitive memory and fractional-order membrane potential dynamics are important and necessary to reproduce neuronal electrical activity.